Pareto-Optimal Nash Equilibria Are Competitive InaRepeated … · 2004-10-12 · JOURNAL OFECONOMIC THEORY 28,320-346(1982) Pareto-Optimal Nash Equilibria Are Competitive InaRepeated

JOURNAL OF ECONOMIC THEORY 28, 320-346 (1982)

Pareto-Optimal Nash Equilibria Are Competitive

In a Repeated Economy*

MORDECAI KURZ

Institute for Mathematical Studies in the Social Sciences,Fourth Floor, Encina Hall, Stanford University, Stanford, California 94305

AND

SERGIU HART

Department of Statistics, TelcAviv University, Tel-Aviv 69978, Israel

Received November 20, 1981; revised June 22, 1981

Consider a finite exchange economy first as a static, 1 period, economy and thenas a repeated economy over T periods when the utility of each agent is the meanutility over T. A family of strategic games is defined via a set of six generalproperties the most distinct of which is the ability of agents to move commoditiesforward in time. Now consider Pareto optimal allocations in the T period economywhich are also Nash equilibria in this family of strategic games. We prove that as Tbecomes large this set converges to the set of competitive utility allocations in theone period economy. The key idea is that a repetition of the economy when agentscan move commodities forward in the time acts as a convexification of the set ofindividually feasible outcomes for player i holding all other strategies fixed. Journalof Economic Literature Classification Numbers: 021, 022.

1. STATIC THEORY DERIVED FROM DYNAMIC CONSIDERATIONS

In "Yalue and Capital" [4, p. 115] Sir] ohn Hicks states the classicaldistinction between statics and dynamics: Statics is timeless while dynamicsinvolves time. This sharp distinction was deemphasized by the extensivedevelopments of dynamic economic theory in the post-war period which,among other things, sought to attain consistency between these twoapproaches. Part of the process of establishing the compatibility of dynamics

'" This work was supported in part by the National Science Foundation Grant SOC75-21820-AO J at the Institute for Mathematical Studies in the Social Sciences, StanfordUniversity, and by the Institute for Advanced Studies at thy Hebrew University, Jerusalem.

3200022-0531/82/060320-27$02.00/0Copyright @ 1982 by Academic Press, Inc.All rights of reproduction in any form reserved.

PARETO OPTIMAL NASH EQUILIBRIA 321

with statics entails the recognition that there are many static phenomenawhich either vanish or change completely when placed in a dynamic context.An example from economic theory which comes to mind and in which sucha situation arises is the difference in conclusions between temporaryequilibrium and full Walrasian equilibrium. Another example is the behaviorof an exchange economy with a storage technology but with or withoutfutures markets.

In the context of game theory this analysis revolved around theconclusions drawn from a single game as opposed to the supergame~theinfinite repetition of the single game. It has long been recognized that thetransition from a game to the supergame enabled behavior to transit fromnon-cooperative to cooperative mode. The celebrated "Prisoner's Dilemma"of non-cooperative behavior is resolved in the supergame through the processof the adoption of essentially a cooperative equilibrium strategy by allplayers (see Luce and Raiffa [10, pp. 97-102]). Along the same line it haslong been known~as a "folk theorem" (e.g., see Hart [3D--that for anygame the set of equilibria in the supergame coincides with the set of allcooperative payoffs (in the single game) which are individually rational. Thisinteresting theorem shows that as we move from the "one-shot" to the"repeated" game, the set of equilibria moves from the narrow set ofnoncooperative outcomes to a set of cooperative outcomes, which is toolarge to provide a definitive theory' of behavior.

In subsequent work Aumann [1] investigated the relationship between thecore of a game and the set of equilibria in the supergame. He shows that theset of all strong equilibrium payoffs in the supergame is identical to the p-core of the single game. With~this same machinery, Kurz [7,8] examines thetheory of altruistic behavior. When considered in the context of a singlegame no altruistic behavior is exhibited yet an extensive range of suchbehavior is possible in the supergame. In a separate paper [9], Kurz studiesthe process of inflation with the same analytical tools.

For some, these considerations may only be a reflection of the deeper prin-ciple that says that the foundation of every cooperative outcome is anequilibrium of a non-cooperative game (e.g., Nash [11]). Yet we adopt theview that it is the interaction of the non-cooperative set-up with the dynamicstructure of the economy which leads to the emergence of cooperativebehavior.

With these ideas as a background we explore in this paper the effect ofrepeating an economy g, a finite or infinite number of times, on the outcomeof the implied game. The reader may note immediately the possible relationto the Debreu-Scarf [2] replica economy. The difference is fundamental: inthe Debreu-Scarf theorem the core converges to the set of competitiveequilibria ,as the static economy is enlarged by replication. We, in this paper,keep the size of the economy fixed but allow it to repeat over time, where the

322 KURZ AND HART

utility of an agent is the mean of his utilities in all periods. To underscorethis fundamental difference note that in the Debreu-Scarf replica economythe convergence of the core to the competitive equilibrium occurs due to therising trading possibilities among the growing number of agents includingtrades which involve identical agents. In our economy which repeats overtime agents do not have increased possibilities of trading with new agents buthave the option of reallocating their consumption bundles over time.Whereas in the replication model, domination occurs with some of thetraders of the same type getting better off and the others worse off, in ourrepeated model, this will not be acceptable unless the average utility over allperiods is increased.

Because of this fundamental difference between replication and repetitionover time, our Theorem 1 below will show that the (utility) core allocationsin the economy which repeats for T periods remains identically the same asthe core of the single period game and thus no convergence of the core of theT period game (denoted CORE(g'T)) occurs. With this result in mind weshift our attention to the set of points which are both Pareto optimal andNash equilibrium allocations in the T period repeated economy and showthat due to the repetition of the economy over time, it is this set whichconverges to the set of competitive equilibria in the single period economy.The key result of this paper can thus be stated as follows:

Consider a standard exchange economy which repeats for T many times and inwhich agents can carry their commodity bundles forward in time. Then the set ofcommodity or utility allocations, which are both Pareto optimal and Nash equilibriain the T period economy converges, as T ~ CD, to the set of commodity or utilitycompetitive allocations in the single period economy.

The crux of the issue at hand is the assumption that agents can' carry theircommodity bundles forward in time and thus we shall be assuming thepossibility of storage. The reader may note that it is this unique and simpleassumption combined with the repetition of the economy over time that leadsto the convergence to the competitive. allocations. There are many recentpapers which showed that various one period games, give, as a solution, theset of competitive allocations. What we show here is that the very simpleassumptions of the existence of storage possibilities and repetition over timeyield the competitive solution. To comment further on these results weemphasize that we treat the repeated economy as a repeated game with a"zero memory," thus establishing an independence over time in the strategieswhich agents can adopt. The set of Pareto optimal allocations in the Tperiod supergame is known to be large. Moreover, we also know that the setof Nash equilibria in the T period repeated game with zero memory is alsolarge. Yet we present here the surprising result that the intersection of thesetwo sets converges to the set of competitive allocations of the single, static,economy.


2. THE ECONOMY

We start with a finite economy g' which is defined byg' =={(Xh, Uh' Wh)'h = 1,2,..., H}, where

Xh = the consumption set of agenth,

Uh= the utility function of agent h defined overconsumption bundles chin X h ,

Wh = the endowment vector of agent h,

and we shall make the following assumptions:

ASSUMPTION 2.

ASSUMPTION 3.

Xh = IR~ for all h.

Uh are monotonic for all h.

ASSUMPTION 1.

ASSUMPTION 4.

Uhare continuous for all h.

Uh are strictly concavefor all h.

ASSUMPTION5. Each individually rational Pareto optimal allocationC = (CI , Cz,..., CIl) satisfies, for all h,

(i) Ch~ 0,

(ii) there is a unique supporting hyperplane to the set {x E IR~ I

Uh(X) > Uh(Ch)} at Ch'

Assumptions 1-5 are not unusual in this type of analysis; conditions thatwould imply them could be given. For example, Assumption 5 will besatisfied if for all h, Uh(') is differentiable in the interior of IR/+, it has infinitederivatives on the boundary of IR~, and Wh '* O.

We denote by cOthe aggregate supply of commodities in g', thus (recallAssumption 5(i»

Il

cO= L Wh ~ O.h=1

(2.1)

Now define (c" cz,..., cH) to be a consumption allocation if r.:= I ch ~ W.Corresponding to every consumption allocation we have the utility allocation(U1(C1),uz(cz),..., UH(CH».With these we use the notation

COREG(g') = the set of all g' core commodity allocations,

CORE(g') ~ the set of all g' core utility allocations,

WG(g') = the set of all g' competitive commodity allocations,

W(g') = the set of all g' competitive utility allocations.

324 KURZ AND HART

We turn now to the economy which is repeated T times and which wedenote by g1'. In this economy we have again H agents with indexh = 1,2,..., H, and the consumption set of each agent is (rR/+r. Being arepeated economy each agent has the repeated endowment vector

(W", w" ,..., w,,) (T times).

For any sequence of consumptions cJ: = (cL cL..., cD of agent h withC~lE R ~I- all t, the utility level of h is defined by

1'1' 1;' (

U,,(c,,)=-.iJ

u,,(c,,).T (=1

(2.2)

As we noted in the previous section, the key assumption of this paper ismade by allowing agents to carry their commodity bundles forward in timein order to rearrange their consumption plans. Thus, although [5'1'should beviewed as a repetition of g over T periods, there is a basic intertemporalstructure of allocations in g1' which is induced by this assumption. This issummarized in Assumption 6 which we state now. To do this we need tointroduce the basic notations of the T period economy:

W~l = total commodity bundle in the possession of h at theend of period t (the amount in "storage"),

z~ = net trade of agent h at time [(positivecoordinates for purchases),

d, = consumption bundle of agent h at time t.

We shall use the following notations:(

((( (

)x = XPX2""'X",

t ( 1 2 ()X=X,X,...,X,

(

( I 2 ()x"=x",x",...,x,,,

with x standing for anyone of W, z, c, and so on.We can now introduce our Assumption 6:

ASSUMPTION 6. Intertemporal Links. The feasible allocations in g1'

consist 01 all (W1',ZT,C1')= {(W~,z~,C~)~~=I};=1 such that for all t

(i)

(ii)

(iii)

(iv)

W~l= W~l-I+ w" + Z~l- cL where we definec~,) 0,W~l)O,\;'/1 (

0~h=IZh= .

W~=O,

PARETO OPTIMAL NASH EQUILIB'RIA 325

The conditions (i}-(iii) can be viewed as conditions of individualfeasibility while (iv) relates to aggregate exchange feasibility. Since in all ofthis paper we are concerned only with Pareto-optimality in gT we do notmake separate use of conditions (i}-(iii) and (iv) and for this reason they arecombined here together.' .

We note that a consumption stream cT = (c1, c2,..., cT) is feasible in gT

(i.e., there are WT and ZT such that (WT, ZT, CT) satisfies Assumption 6) ifand only if

( H

L L c~<troT= I h= I

foralll<t<T. (2.3)

In the proofs below we use (2.3) rather than Assumption 6 directly.This completes the formal description of gr. However, before proceeding

to our analysis we can show immediately that Assumption 6 does not changethe set of Pareto optimal allocations that were not in g T withoutAssumption 6. To see this note that for any Pareto optimalCT=(C1,C2,...,CT) in g'T, the. strict concavity Assumption4 ensures(Lemma 1 below) that c( = c = constant vector for all t. But this means thatcommodities are not moved forward in any Pareto optimal allocation in g'T

and no use is made of Assumption 6.We now turn to CORE(g'T) and ask if Assumption 6 combined with the

repetition over time induces any convergence of this core. For this, we definefeasibility for every coalition S c {I,..., H} using Assumption 6 with

LhES z~ = 0 as (iv). With this definition we now have

THEOREM 1. Let Assumptions 1-6 hold, then

(i)

(ii)

c E COREG(g') if and only if cT = (c, c,..., c) E COREG(g'T),CORE(g') = CORE(gT).

Theorem 1 states that the core of g'T consists precisely of the stationary

allocations in g'T, each one being an identical repetition of a core allocationin g'. We note that Theorem 1 is also true without Assumption 6 (e.g., whenfeasibility is required for each period t separately, or commodities may bemoved both forward and. backward in time, and so on) and thus thepossibility of rearranging consumption by moving commodities forward intime has no effect on the core of g'T.

In relation to the Debreu-Scarf [2] replica economy, Theorem 1 showsthat the T repetition of the economy does not alter the core utility andcommodity allocations. Thus the idea of attaining the convergence ofCORE(g'T) in the repetitive economy in an analogous way to theconvergence of the core in the replica economy, must be abandoned.

The objective of this paper is to show that when the simple assumption of

326 KURZ AND HART.

,

moving commodities forward in time (Assumption 6) is made, the set of allNash equilibria and Pareto optima in g'T converges to W(g'). But to do this

we must now turn to a description of our strategic considerations relative towhich one can define the concept of Nash equilibrium.

3. A FAMILY OF FINITE STRATEGIC GAMES"

In order to talk about Nash equilibrium in g'T we need to talk about astrategic game. However, Theorems 2 and 3 which are the main object ofthis analysis remain true for a broad class of strategic games as long as theyhave certain properties to be discussed now. Thus we define F(g'T) to be aclass of all the strategic games in gT, with the following properties:

P.I PLAYERS. The set of players is{ 1, 2,..., H}.

P.2 STRATEGIES. The set of strategies of player h is Sr.

From now on, we will denote by or elements of Sr,oT = (oi, oi,..., o~),d T ( 1' I' T 'T )an O(h) = °1"'" O"-I'°h+I"'" °H .

P.3 COMMODITY OUTCOMES, Each H-tuple of strategies 01' results, in acommodity outcome (WI', ZT, CT) which satisfies

(i) Assumption 6 (feasibility),

(ii) -Z~l -< Wh for all h all t (zero memory).

Moreover, for every (WT, ZT, cT) satisfying (i) and (ii), there exists an H-tuple of strategies whose commodity outcome is precisely (WI', ZT, CT) (com-prehensiveness ).

PA PAYOFFS. For each H-tuple of strategies 01', the payoff of player hIS

1 I'

- L u,,(c~),T 1=1

where (WI', ZT, CT) is the corresponding commodity outcome.

P.5 INDIVIDUALSTRATEGICOPTIONS. Let 01' be an H-tuple of strategies,resulting in the commodity outcome (WI', ZT, cT). Then, for each h:

(a) Forward Reallocation. For each t < T and v E [R/+ with v -< c~,there exists a strategy ar E Sr of player h, such that the commodity outcome

~T"7"'T . I' "I' .(W , z , c ) correspondmg to (0 (h)" o h) satisfies"


iT=ZT,

"1' l'Ck = Ck andA

l'A

l'Wk=Wk, for all k =1=h,

"T TCh = Ch and Wh =~,

and W~ = W~ + v,

and W~+I = W~+I.

for all ! =t=t, t + 1,"I ICh=Ch-V

C~+I = C~+I + v

,Note that a repeated application of P.5(a) for varying t, enables the playerh to reallocate his consumption! forward in time, namely, to replace cr byany cr that satisfies L~= I C;, ~ L~=I c;, for all t.

(b) Option to Refuse Trading. For each sequence tl, t2,..., t;/of dates,there exists a strategy ar of- player h, such that the commodity outcome(WT, iT, CT) corresponding to (CJfhPon satisfies

"I 0Zh =- Z'- h

if t=tjforsomej= 1,...,J,

otherwise.

We remark immediately that the possibility of refusing to trade at time tdoes not violate the feasibility of trades at all other dates due to P.3(ii). Thusone needs to think of P.3(ii) and P.5(b) together. It is clear that this is not aninconsequential assumption. However,' it should be pointed out that in thefinite repetition case we use it only for stationary commodity outcomeswhich do not require any intertemporal transfers of goods (see Proof ofTheorem 2 and Lemma 1). Thus, in this case a change in trade in someperiod cannot affect the feasibility of trades in the other periods. In thissense, the assumption we actually use is to be viewed as much weaker thanthe combination of P.3(ii) and P.5(b).

P.6 RESPECT FOR PRIVATE OWNERSHIP. Let (WT, ZT, cT) be acommodity outcome. Then every player h has a strategy ar such that

(i) the commodity outcome resulting from aT = (a;, ar,..., aJ;) is(WT ZT CT ), , ,

(ii) for every player k and every strategy or. the commodity outcome(\\,1', iT, CT) corresponding to (a;k)! oD satisfies, for all t, eitherl i~ - z~ or"I 0Zk = .

I "either... or ..." refers to each ( separately, i.e., for some ('S, zi = z i, and for the others,

i~= O.,-

328 KURZ AND HART

Discussion

Although not explicitly stated the condition of "zero memory" is at thebasis of the above properties. Simply stated this condition means that nostrategy can be defined as a function of past events. It implies that playerscannot threaten to retaliate in the "future" as a result of "past" or "present"deviations. We do not assume here that such retaliations are prohibited butrather specify in P.5-P.6 those individual actions which cannot lead toretaliations.

The idea of "zero memory" also appears in property P.3(ii). To see thisnote that due to zero memory the trades at times t cannot rely upon the.actual occurrence of exchange or storage activities prior to time t, and thus itis only natural that as an outcome condition for time t no individualexchange cart' exceed the available resources of an agent at that period (thisis condition P.3(ii)). This is also related to an implicit demand that eachplayer accumulates commodities for consumption only, and thus if he wishesto obtain a commodity for later consumption he should obtain it early andcarry it by himself.

Conditions P.3, P.5(a) and P.5(b) must be understood together. On theone hand, any commodity outcQme (WT, ZT,CT) can be realized by somestrategy aT. In this sense the strategy aT realizes the exchanges z~ in [fT. Onthe other hand, player h may deviate from aT under two special situations:

(a) He is able to move commodities forward in time (for consumptionor storage purposes), provided he already owns these commodities under theoutcome of aT,

.

(b) He may refuse to exchange any commodity at all in a specified listof dates.

The idea of "respect for private ownership" is brought into the model sothat no player can force other players to yield to him any portion of theirendowment if they do not wish to do so. This suggests, for example, that noplayer can use a threat strategy which' will tax away part of the endowmentof. other players. These icieas are expressed by the property that relat,ive toany feasible trade outcome ZT the players can adopt a type of strategy whichwe denote by aT that ensures that a deviation by any player h means one oftwo things: either h trades according to ZT or he does not trade ~t all. Notethat we do not make this assumption for all aT---'but, rather, that to eachcommodity outcome corresponds at least one H-tuple of strategies thatrespect private ownership.

A comparative reference must now be made to a growing literature on thestrategic approach to Walras equilibrium (e.g., Hurwicz [5, 6], Schmeidler113] and many others). In this literature the market economy is charac-terized by"a game in which the strategies of the players are messages and the


outcomes are allocations. It is then proved that the outcomes of Paretooptimal Nash equilibria coincide with the Walrasian correspondence. Theproblem, however, is that the family of mechanisms which generate theabove result is rather limited. For example, in Hurwicz [5] and Schmeidler[13] the strategies of the players. include price messages so that given theprices proposed by others a player can then select from a feasible budget set.In recognizing this limitation Hurwicz [6] finds it necessary to impose thecondition that, holding the strategies of all other players fixed, the set offeasible outcomes for any given player is convex. This obviously generalizesthe idea of price messages for trading purposes. .

The fundamental idea of our paper is that the large family of gamesdefined here do not require the convexity of individual feasible outcome setssince repetition of the economy enables us to derive this convexity as aconclusion. We regard the convexification which results from repetition to bean interesting result which could have important implications elsewhere.

An Example of a Game in r(g'T)

Define a strategy a rof player h to consist of two parts:

(i) A "proposed" sequence ~r of net trades with -,~ ~ Wh all t.(ii) A "proposed" sequence of consumptions 'Yr.

The commodity outcome is then defined as follows:

I ylZh = "'h if

H

L ,~= 0h=l

=0 otherwise,t t

Ch = Yh if W t-l I + '""-t

h +Zh WhPYh'

=0 otherwise,

Wt= Wt-l +Z~+Wh-C~,

It is easy to check that all the assumptions and properties specified above aresatisfied; proposals for trade materialize only when unanimous agreementtakes place and each trader has complete control both over his storage (thusover his intertemporal transfers of commodities) as well as over hisconsumption-as long as they are feasible.

Having defined the family r(g'T) we shall now proceed. to use theterminology "Nash equilibrium in g'T" to mean "Nash equilibrium in anygame Y in the family r(g' T) which has the five properties P.I-P.5." Althoughthe set of Nash equilibria of each particular game Y depends upon the

. specific definition of each such game, our results, concerning the commodityand utility outcomes, hold for any game Y in the .Camily r(g'T).

330 KURZ AND HART

For each feasible allocation in gT define the average consumptions aT =(aJ",af,..., a~) and average utilities UT = (Uf, uf,..., U~) as follows

I'1 {, t

ah=-kch'T t=1

1 I'Uf:(CT) = T .L Uh(C~).

t= I

Now denote by

PO T- the set of all average utilities UT = (Ui, uf,..., Un

associated with Pareto optimal outcomes (WT, ZT, CT) in gr.

PO~ = th~ set of all average <;onsumptions aT = (af, ar,..., a~) "associated with Pareto-optimal outcomes in gr.

NE T = the set of all average utilities UT associated withNash Equilibria in r(g'T).

NE~ = the set of all average consumptions aT associated with

Nash Equilibria in r(g'T).

We can now state our first main result:

THEOREM 2.then

Let Assumptions 1-6 and Properties P.I-P.6 be satisfied

(i) W(g) = limT->co [POT n NET],

(ii) WoCg') = limT->co[PO~n NE~],(iii) for every Pareto-optimal outcome. (WI', ZT, CT),

I 2 I'C =c .='..=c.

Theorem 2 is the first of our two main convergence theorems. The setPO Tn NE T is the set of Pareto optimal utility allocations in g T which arealso Nash equilibria in F(g T). The theorem suggests that the limit of this setis the set of competitive utility allocations in g. Similarly, the setPO ~ n NE~ is a set of mean commodity allocations in the T periodeconomy and this set converges to the set of commodity allocations Wo(g).

Theorem 2 formalizes our earlier claim that if we start with any non-cooperative behavior in a single economy then the repetitive economy withthe property specified in Assumption 6 will undergo a transformation into aneconomy with a more specialized behavior leading to an outcome defined bythe set of competitive equilibria in the one period game. Alternatively one


can interpret the result of Theorem 2 to mean that a competitive behavior inthe one period economy will be strongly reinforced by the repetitive nature ofthe economy and the ability of agents to transfer commodity bundlesforward in time.

We now motivate Theorem 2 with the aid of the familiar Edgeworth box.Consider an economy with individuals I and II. In Fig. 1, E is thecompetitive equilibrium while A is a Pareto optimal allocation on thecontract curve. Since the indifference curves are strictly convex a Paretooptimal allocation A which is not a competitive equilibrium has the propertythat the line wA contains points like B = (xf, xfl) which from the point ofview of I, are strictly better than A (thus UI < uf)o Such points satisfy thecondition that for some 0 < A. < 1

xf ~ AWI + (1 - A) x1.

We shall show that the repetition of the economy provides individual I astrategy to attain convex combinations like xf and thus reject xA as apotential Nash Equilibrium in r(g'T). To see this suppose A.= 1/4 and it isproposeq to trade w into xA for T periods. Table I shows how individual Ican alternate his decisions over time to particupate and trade into xA or notparticipate and stay at WI and in so doing attain xf.

The table shows that at t = 0 the individual consumes nothing, but in allsubsequent periods he consumes exactly xf=(1/4)wl+(3/4)x1, and thishe does by not trading every fourth period. As the length of T increases theinitial loss at t = 0 becomes smaller and disappears in the limit. Also if A. isnot a rational number the approximation becomes more accurate as T

N

>-f-a0~:E0u

IT

ICOMMODITY 1

FIGURE 1

ResourcesA vailable in t

Decision (endowmenttaken + trade) Consumption

Do not trade WI 0

Trade x~ I 3 A;jWI + ;jXI

Trade x~I 3 A;jWI + ;jXI

Trade x~ lw + lxA4 J 4 I

Do not Trade 1 3 AWI ;jWI + ;jX1

Trade x1I

'+3 A

.WI ;jXI

Trade x~ tw( + ~x~

332 KURZ AND HART

TABLE I

Decision Table to Attain x: = (1/4) WI + (3/4) x~

Time

An alternative program

Proposedprogram of

consumption

0

1

2

3

4

5

6

X~

x~

x~

x~

x~

x~

x1

Storage

WIJ I A;jWI + ;jX,

2 2 A;jWI + ;jXI

I 3 A.WI + ;jXI

WI

J I A;jWI + ;jXI

2 + 2 A;jWI' .X.

becomes large and A is approximated by the fraction of the total number ofperiods in T during which the agent does not trade. It follows that for largeT the Pareto optimal allocations which cannot be improved upon with theconvexifying strategy are allocations like E-the competitive ones.

4. THE INFINITE REPETITION

Our object now is to extend the analysis to goo, the infinitely repeatedeconomy. Assumptions 1-6 and P~operties P.I-P.6 are carried over, the QpJYchange being that the feasible commodity outcomes are now infinite streams;using the notation x (rather than x 00) for (x!, X2 ,..., Xl,...), they will bedenoted by (W, z, c).

Unfortunately, this extension is not a straightforward procedure sincethere are two basic difficulties. The first relates to the problem of defining"domination" and thus Pareto optimum in goo and Nash equilibrium in theinfinitely repeated games F(g'oo). The second relates to the problem ofdifferentiating among different exchange patterns all leading to essentially thesame outcome.

'

To clarify the first difficulty consider any feasible allocation (W, z, c) ingoo. Define the terms -

T 1 ~ tah = - 2.., Ch'

T l=!

T 1 {~ t)Uh(Ch) = - L Uh(Ch .T t=!

(4.1 )

(4.2)

TABLE II

Commodity outcome A Commodity outcome B

Time ell z~ W~ e~ z~ W~

I (0,0) (0,0) (0,2) ( I, I) (I, -I) (0,0)

2 (I, I) (2, -2) (1, 1) (1, 1) (1, -1) (0,0)

2 (1, I) (0,0) (0,2) (1, I) (1, -1) (0,0)

4 (1, 1) (2, -2L (1, 1) (1, 1) (1, -1) (0,0)

5 (1, 1) (0,0) (0,2) (1, 1) (1, -1) (0,0)

6 (1, I) (2, -2) (1, I) (1, 1) (1, -1) (0,0)


Now, if we wish to compare c with any other feasible consumptionallocation c we may have a difficulty due to the fact that the sequencesjUk(Ch)}r=l and {UnCh)}r=l may not converge. In order to define apreference relation for the comparison of Ch and Ch' we introduce twodomination relations discussed by Aumann [1]. The two dominationrelations called >-u (upper) and >-L (lower) are defined as follows.

Let a = jat I~ 1 and 13= {Pt} ~ .be two infinite sequences.

DEFINITION4.1 a. The sequence a is said to Upper-dominate thesequence 13(denoted a >-u13) if there exists an infinite sequence of dates tj,j = 1, 2, 3,... and e > 0 such that

at} > Pt} + e, j = 1, 2,3,....

DEFINITION4.1b. The sequence a is said to Lower-dominate thesequence 13(denoted a >-L13)if there exists a To and e > 0 such that

at > Pt + (; for all t > To'

With these two definitions of domination w~ have two induced concepts ofPareto optimum, and two induced concepts of Nash equilibrium. Next, let us

,address the second problem mentioned above. To do this we 'start with asimple example (see Table II).

Consider an economy with two agents called 1 and 2 and twocommodities. The two endowments are WI = (0, 2) and W2= (2, 0). Assumethat the allocation [(1, 1), (1,1)] is Pareto optimal in g. Now consider inTable II two different commodity outcomes in goo. Clearly c~ is the same inA as in B except that in Bagent 1 consumes (1, 1) in the first period while

334 KURZ AND HART

he consumes (0,0) in A. However, by our standard of comparison in goo,

the consumption sequences in A and B will be equivalent. Yet the exchangepattern in A leaves the agent much less freedom than in B. This is so since inA the agent needs to maintain the storage strategy and if he wanted to alterc~ so as to consume (1, 1) in every second period and (0, 2) at the rest of thetime points-he cannot do so. In situation B the agent is free of the storagerestrictions and is free to alter his consumption pattern if he wished.

The critical difference between A and B is found in the useless storageactivity which is carried out in A and our aim is to insist that in a Paretooptimal program, the agents do not carry out unneeded storage. Clearly, oneway of doing so is the introduction of a direct storage activity which wouldconsume resources but this will complicate the present model and willobscure the main results. An alternative way of removing useless storage isby introducing a fiction in the form of a vector q E IR~ of "theoretical"storage cost per unit of commodity and per unit of time. This means that if(W, z, c) is a commodity outcome in goo, then we shall define the q-utilityoutcome up to time T to be

T 1 ~[ t t ]U/t(c/t,q)=-~ u/t(c/t)-q,W/t.

T [=1(4.3)

We denote the infinite sequence of utilities by

U/t(c/t, q) = {uh'(c/t, q)}~=I' (4.4 )

Now, for any q we can compare c with c using the definition (4.4) and thetwo domination relations. in Definitions 4.1.a and 4.1 b; This procedure willlead to a concept of Pareto optim.um which will depend upon q~a resultwhich we cannot accept since q is viewed as fictitious storage cost intendedto remove all unneeded storage. Since we do not want the cast of storage tobecome an essential part of the payoff we shall let q decrease to 0 and seekthe limiting behavior. The following will make these ideas precise.

DEFINITION 4.2. Let q ~ O. A commodity outcome (W, z, c) in goo islower Pareto optimal with respect to q if

(i) there is no commodity outcome CW,z, c) such that

U,Jc/t, q) > U/t(c/t, q)L

for all 11;

(ii) for all 11,the sequence {U;;(c/t, q)}~=1 has a limit ("summability").

We remark immediately that the summability condition (ii) is not relatedat all to Pareto optimality; we include it in the definition so as to belable todefine "payoffs" (see also Aumann [1 D.

PARETO OPTIMAL NAS~ EQUILIBRIA 335

Let us denoteI, r

POL(q)=the set of all limits of average utilities (i.e.,{UrcCh,q)}r=l-see (4.3» for L-Pareto optimaloutcomes with respect to q.

POGL(q)= the set of all limits of average consumptions -(Le.,{an r;!=I-see (4.1» for L-Pareto optimal outcomeswith respect to q. .

In a similar manner one defines the notion of Upper Pareto optimalitywith respect to q, and the corresponding sets POu(q) and POGu(q).

We now turn to the issues of Nash equilibrium. The class of strategicgames r(gOO) is assumed to satisfy the properties P.I-P.6 as do the games in

T . .

reg ) (I.e., take T = 00).

DEFINITION4.3. Let q ~ O. An H-tuple of strategies 0 resulting in thecommodity outcome (W, z, c) is a lower Nash equilibrium with respect to qif

(i) for all h, there is no strategy Oh of player h such that thecommodity outcome (W, z, c) corresponding to (O(hl' °h) satisfies

Uh(Ch, q) >- Uh(Ch' q),L

(ii) for all h, the sequence {UrcCh' q)}~=1 has a limit ("summability").

We now denote

NEL(q) = the set of all limits of average utilities for L-Nashequilibrium with respect to q

NEGL(q) = the set of all limits of average consumptions for L-N ash equilibria with respect to q.

Again by replacing "lower" with "upper" we obtain the sets NEu(q) andNEGu(q). .

It should be pointed out that, as q decreases to 0, the sets POL(q) need nothave a limit; and the same is true for NEL(q) and for the other sets which wedefined above. Let us denote'by limq->oH the limit as q ~ 0 and q ~ O. Ourresult is

THEOREM3. Let Assumptions 1-'6 and Properties P.I-P.6 hold for goo

and r(gOO). Then

(i) W(g) = limq ->o++[POL(q) n NEL(q)] = limq->oH[POu(q)nNEu(q)], .

336 KURZ AND HART

(ii)NEGU(q)],

(iii) for any q ~ 0 and every allocation (W, z, c) in goo which is L orU-Pareto optimal wih respect to q, the sequence a = {aT}~=1 converges.

Theorem 3 provides the natural extension of Theorem 2 to the infiniteeconomy goo. It says that in goo the set of all Pareto optimal allocationswhich are also Nash equilibrium in r(gOO) have the same character as thecompetitive allocations in g. This means that the utility limits of all such care exactly the utility allocations in W(g) and the limit mean consumptionallocations aT are exactly WGUf).

WG(g) = limq--.o++[POGL(q) n NEGL(q)] = limq--.o++[POGu(q)n

5. A FINAL NOTE

As indicated in Section I the present paper is part of a general search ofnon-cooperative mechanisms which tend to change their nature due torepetition. We think that the extent of such phenomena is more widespreadthan is generally recognized. Our aim is to explore the factors whichgenerate the motivation for the convergence toward- a more cooperative modeof behavior. In the present paper we find that the ability of agents to. carrytheir endowments forward in time combined with the repetition of theeconomy are adequate factors to generate a tendency toward competitiveequilibria.

6. PROOFS IN g T

-.,

LEMMA 1. Let cT E PO(g'T). Then cT is stationary, i.e., there is c suchthat cl = Cfor aliI < t < T. Furthermore, c is Pareto optimal in g', and thereare no transfers of commodities between periods.

Proof Define

I {.1

ah = - L , ChT 1= I

for all h.

The stationary consumption aT = {(a l' a2 ,..., aH)} i= 1 is feasible (and noforwarding of commodities is needed), since

HIT H H

I ah = - I I c~ < I Who

h=l TI=lh=l h=l

(6.1)


By Assumption 4, each Uhis strictly concave and since c T is Pareto optimal,we must have c~ = ah for all t and all h. If b Pareto dominates a in g', thenbT = (b, b,..., b) Pareto dominates aT in g'T; therefore a is Pareto optimal ing'. By monotonicity (Assumption 2) Lh ah = Lh Wh' hence no transfers ofgoods are made between periods.

Proof of Theorem 1. Since every point in CORE(g'T) is also Paretooptimal in g' T, it follows from Lemma 1 that it is stationary in utilities andcommodities. The theorem is now immediate.

LEMMA2. Let x be an allocation in g. Then x is competitive if and onlyif x is individually rational and Pareto optimal in g' and, for all m = 1, 2,...,

(m-l 1

)Uh Xh + - Wh -< Uh(Xh)mm '-

for all h. (6.2)

Proof. If (p, x) is a competitive equilibrium in g', then x is clearlyindividually rational an~ Pareto optimal, and

(m-l 1

)p .

mx h + m Wh-< p . W h ,

implying (6.2)., Conversely, if x isUh(y) >Uh(Xh)} fromAssumption 2, p >O.

From (6.2) and the concavity ofuh' we obtain Yhn{axh+(l-a)whI 0-<

a ~ 1}= 0. We can thus separate these two sets by a vector p k E IRI.However, since Ph also separates Xh and Yh, and since by Assumption 5(ii)such a separating hyperplane is unique, it follows that Ph may be takenwithout loss of generality to be Pk = p. But then P . Wh -<p . y for all y E Yh,

which implies P . Wh -<P . Xh' This holds for all h; together with Lh Wh =Lh Xh' and P >0, we get p . Wh = P . Xh for all h, completing our proof.

Remark. We actually proved that in the statement of Lemma 2 we couldhave concluded that uh(axh + (1 - a) Wh) ~ Uh(Xh) for all 0 ~ a ~ 1.

Proof of Theorem 2. We prove the theorem in two steps:

Step 1. limT-->oo[POTn NET] c W(g'),

Step 2. POT n NET ~ W(g').

Step 1. limT~oo[POT n NET] c W(g'). Let aT be an H-tuple ofstrategies with a commodity outcome (WT, ZT, cT) such that cT is Paretooptimal in g'T and aT is a Nash equilibrium in some y E F(g'T). By Lemma 1c T is stationary (Le., ct = c for all t), no commodities are transferred forward

Pareto optimal, let p E IRI separate Y h = {y E IRItI

Xh' for all h; such a p clearly exists, and by

Total wealth Net trade W~at! - 1 at at W~-l + w" ( Wt-I=

"+ w"

(W~- 1) (z~) + z~ t + z~- c~)C"

I 0 0 w" 0 w"2 w" + c"

1 2 2 1W" C" -' W" jW" + jC" jW" + jC"

3 2 1 2 4 I 2 1 2}w" + }C" c" - W" }W" + jC" jW" + }c" }w" + }c"

4 lW t-2,

0 4 2 1 2} ,,- }c" }w" + }c" }w" + }c" w"5 w" c"-w,,

338 KURZ 'AND HART

and e is Pareto optimal in fJ'. By P.5(b) and P.5(a); player h may'refuse totrade at all dates, and consume his endowment Wh in each period. Since 01' isa Nash equilibrium we conclude that e is an .individually rational Paretooptimal allocation in g -hence we can use Lemma 2.

.

Now let player h use P.5(b) as follows: let m be a positive integer, andconsider the following new strategy of h: in each of the dates t - 1, m + 1,2m + 1,..., km + 1,..., he does not trade (i.e., i~ = 0), and in the othersi~l= z~. His consumption will be Ch= 0 and c~ = «m - 1)lm) Ch+ (1/m)whfor all t > 1. We claim that this consumption plan is feasible for h (by use ofP.5(a»); indeed, consider the first m + 1 periods~ At t = 1, h has w", wVich isconsumed in 'the next m periods (at the rate of (11m) w" each); fort = 2,..., m, only «m - 1)lm) e" is consumed, the remaining (11m) c" beingstored in order to be able to consume « m- 1)1m) e" in period m + 1 (whenthere is no trade again). Therefore h can get [«m - 1)lm) Ch] + (11m)w" in

. each of the periods 2,..., m + 1 by usingw" from period 1 and e" from

periods 2,..., m. Furthermore, at the start of the next cycle, at t = m + 2, hhas again w" (stored from t = m + 1), so that the whole process may berepeated. (See Table III for an example.) The feasibility follows' from thisconstruction. .

The utility to h of {c~}i=1 is

~-U,,(o) + T;1

u" (m:1

c,,+ ~ w,,),

Since 01' is a Nash equilibrium, we must have

~u,,(O)+ T;

1u" (m:

1c" + ~ W,,) <u,,(e,,). (6.3 )

TABLE III

Example of the Use of Property P,5(a): m = 3


As T -+ 00, we get the inequality (6.2). The same construction may be donefor all m = 1,2,..., and for all h. Lemma 2 therefore implies that c E wdtW)and (UI(CI)' U2(C2)"'" UH(CH» E W(tw).

Step 2. W(tw) c (POT n NET). Let C E Wo<g) and consider thestationary commodity outcome (WT, ZT, cT) given by c~ = Ch' z~ = Ch - Wh

and W~ = O. It clearly satisfies P.3(i) and (ii) and by comprehensivenessthere is aT resulting in it. Moreover, by P.6, we may choose aT so as torespect private ownership for all h. Assume now that some player h deviatesand by P.6 his trades will necessatily be either Ch- Wh or O.

By concavity of Uh the average utility of h will be' at most

(TI T - TI

)uhTwh + TCh,

where TI is the number of periods in which h's trade was O. By Lemma 2(see remark at the end of the proof) CE WG(tw) implies that

(TI T - TI

)Uh T Wh + T Ch ~ Uh(Ch)'

,thus aT is a Nash Equilibrium. As for Pareto optimality in tWT it followsdirectly from the Pareto optimality of any Walrasian allocation in g and thestationarity of cT.

It is interesting to question the need for the assumption of strict concavity(Assumption 4). It was used in an essential way in Lemma 2 above and theexample to be presented now will show that our results will not hold withoutit.

EXAMPLE. Let 1=2, H = 2, WI = (1,0), W2 = (0,1), and a commonutility function for h = 1,2 that is Uh(X,y) = (Vi + y'y)2. This is a concavebut not strictly concave utility function as it can be seen that along any rayfrom the origin the function is linear. Now consider the following plan:

For l = 1,2,..., T - 1: no trade and no consumption. For l = T:c: = «T/3), (T/3n,ci = «2T/3), (2T/3)) and the corresponding trade at an exchange rate of 2/3 ofCommodity 1 for 1/3 of Commodity 2.

This plan is Pareto optimal in tWT since (lIT) Li= I uI(cD = uI(1/3, 1/3)and (lIT) Li= I U2(C~) = u2(2/3, 2/3). The allocation is also a Nashequilibrium in gT since none of the agents can improve due to the fact thatthe only available alternative is not to trade at all in period T giving each anaverage lltility of Uh(Wh)' The key observation to make is that the meanutilities UI (1/3, 1/3) and u2(2/3, 2/3) are not a competitive equilibrium in g

340 KURZ AND HART

since the unique competitive price is p* = (1/2, 1/2) and for agent 2 thepoint (2/3, 2/3) is outside his budget set

Bip*) = {xlp* . (0, I)}.

7. PROOFS IN goo

LEMMA 3. Let q ~ 0 and let (W, z, c) be L-Pareto optimal with respectto q. Then there is a E POG(g) such that, for all h,

(i) I'l'1m ah = ah

1'->00.

and for all c5> 0,

lim ~#{t<Tlllc~-ahll<c5}=l,1'->00 T

(ii)

(iii) lim ~ # {t <T III W~ II < c5} = 1,1'->00 T I

(iv) rim ~#{t<Tlllwh+z~-ahll<c5}=1.1'->00 T

Proof. Denote by vh the expression

l'

V h = lim ~ L (Uh(C~) - q .W~).1'->00 T

1= 1

(7.1)

Since the sequence {a1'}r= 1 is bounded (for each h, by a and cO)let a be alimit point of this sequence, The concavity of Uh implies (recall (4.1» that

l' l'l' 1~ 11'--' [ .1 I ]uh(a,J ~ T L , Uh(Ch) ~ T L , Uh(Ch) - q . Wh .

1=1 1=1

The continuity of Uh then implies

uh(ah) >- v h for all h. (7.2)

Consider now the stationary allocation (W, i, c) given by c~ = a h + (1/H)L~=l (wh-ah) and i~I=C~-Wh' This is clearly feasible with W~=O forall h and all t, Since L~;= 1 (coh - all) >- a (by feasibility) it followsfrommonotonicity that the payoff to Chis greater than uh(ah)' Since (W, z, c) is L-Pareto optimal with respect to q, (7.2) implies that in fact

(

I/{

)

,

Uh ah + HL (evh - all) = u~(ah) = Vh,h=l


hence it also follows that L~=l Wh = L~=l ah' Furthermore, a EPOG(g),otherwise, if b dominates it, the' stationary allocation associated with b willL-dominate (W, z, c), hence it will also dominate (W, z, c) (by 7.2).

Let a' be another limit poiht of {aT}~=l' As above, uh(ah)=vh anda' E POG(g); the strict concavity of all Uh implies a = a' (consider(1/2) a + (1/2) a'). Hence the sequence aT must actually converge to a,proving (i).

Next, fix h and let ..t be a super-gradient of Uh at ah (cf. Rockafellar [12,Theorem 23.4, p. 217], and keeping in mind Assumption 5(i».

Define

lex) = uh(ah) - Uh(X)- ..t(ah - x),

for all x E fR~.Thenf(x) ~ 0 for all x, andf(x) = 0 only for x = ah (becauseof the strict concavity of Uh)' Furthermore, f is a convex function, therefore

f(ax + (1 - a) ah) ~ alex) + (1 - a)f(ah) = alex) ~f(x)

for all 0 ~ a ~ 1 and x E IR1+.Hence f does not decrease as x gets "fartheraway" from ah' and we get

inf{f(x)I II x - ah II ~ c5} = inf{f(x)

I II x - ahll = c5}.x x

The latter infimum is attained since f is continuous and c5 > O. Denote thisinfimum by p.

Consider now

1 ~ t t ]uh(ah) - T L.., [Uh(Ch)- q . Wht = 1

= ~ ~I[u.(a.) - u.(c~) -.t. (a. - c~)1

A t(

1 ~ t

)+ . (ah- ah) + q.T t~l

Wh

1 ~ t T

(1 ~ t

)~T /;;If(ch) +A' (ah -ah) + q.T t-21

Wh

~A' (ah-dh)+p. ~#{t~'Tlllc~-ahll~c5}

+ c5. min (q)j' Tl

#!

t <TI t (W~)j ~ bl.l(j(1

. 1~1 ~

342 KURZ AND HART

As T --7 00 the left-hand side of the above inequality converges to

(v" - v,,) = 0 and since A(a" - aD --70 it follows that

1p - #{t ~ T III c~ - alII! >J}

T

I

!

l

!

+ J m~n (q)J' -# t ~ TI L (W~)J >J ~ O.

1<,) <,l.

T 1= 1

However, since p > 0, J> 0 and minI <,J<,l(q)J> 0, conclusions (ii) an.d (Hi)of the lemma follow immediately. To prove (iv) note that

(w" + zD - a" = (c~- a,,)+ W~+1

- W~

and this shows that (iv) follows directly from (H) and (Hi).

Proof of Theorem 3. The definitions of lower and upper dominationimply that for all q, POL(q)::) POu(q) andNEL(q)::) NEu(q), and thus

[POL(q) n NEL(q)] ::) [POu(q) n NEu(q)].

We recall that, for a collection {A(q) }q~O of sets,

lim sup A(q) = n U A(q),q-tO++ - -q~O O<t,q<,q

lim inf A(q) = U n A(q),q-tO++ q~O O<t,q<,q

and the limit exists if and only if lim sup = lim inf.Our proof will therefore consist of the following two steps:

Step 1.

Step 2.

Step 1.

lim SUPq-to++[POL(q) 0 NEL(q)] C W(tw). .

W(tw) c [POu(q) n NEu(q)], for all q ~ O.

Urn SUPq-to++[POL(q)n NEL(q)] c W(tw)~

Let (J be an H-tuple of strategies with (W, z, c) as, commodity outcomesuch that for every q ~ 0 there is 0 <{q ~ q with respect to which (W, z, c) isL-Pareto optimal and L-Nash equilibrium. By Lemma 3, a = limT-tco aT is in

PO d g). This allocation is also individually rational since h can alwaysselect the strategy which yields w" as a stationary consumption and no trade.We have to show that it is a Walrasian equilibrium.

Thus assume a E WG(Jr). By Lemma 2 there are hand m such that

(m - 1 1

)u" m a" + -;;; wIt > u~(a,,).


Define

1[ (

m - 1 1) ]

e =4"" u" m a" + m w" - u,,(a,,) > O. (7.3)

By the continuity of u" (Assumption 3), let b > 0 be such that

(m - 1 1

)x -

mah +

mWh <b

implies

I

u.(x) - u. (m: 1u. + ~ w.)

I

< t. (7.4)

Next, we call a date t "good" if it satisfies

II(W"+z~) - ahll<b, (7.5)

and "bad" otherwise. Let {tjU~ I be the sequeJ1.ce of "good" dates (it isinfinite by (iv) in Lemma 3).

We now apply properties P.5(a) and P.5(b) of the strategic game to obtaina strategy a" of player h according to which he refuses to trade on datestkm+I' k = 0, 1, 2,..., and then move commodities forward so that thecommodity outcome (W, z, c) resulting from (o("P a,,) satisfies the following:

"t 0z"=-zt- "

for t = tkm+ I , k = 0, 1, 2,...,

otherwise,

"t.

tC,,=Wh+Zh

=0

m -1 1= (w+z~)+-w"m m

for t "bad" date, -

for t = t I

for t = tj' j '* km + 1, k = 0, 1, 2,...,

= ~ ( f Z~(k-I)m+i

).+ Wh

m 1=2for t - tkm + I' k - 1, 2,....

The idea here is similar to the one used in the finite case (see Table III).On "bad" dates agent h continues to trade z~ as before and consumes

(w" + z~). Now, not counting "bad" dates, consider only the "good" onest I' tz, t3"'" The agent divides these dates into cycles of length m and at dates

tkm+ l'k = 0, 1, 2,..., he does not trade. He starts ofTthe consumption and

storage program by consuming 0 at tI and- storing w" at that time. Followingthis initial,.step, for each cycle of length m (after the initial date t 1) the agent

344 KURZ AND HART

consumes during each of the first m - 1 dates only (m - 1)Im of the netquantity (wit + z~J and 11m of the initial endowment wit out of storage. Inthe mth date there is no trade and the amount which is in storage at the startof the period is devoted to consumption. The cycle starts again by puttinginto storage (at the end of the mth data) the endowment wit of the period.

We will show that there is To and q ~ 0 such that for all T> To and all0 ~ q <: q.

1 T ,

~ I~

I, - L, [UIt(CIt) - qWIt] > ult(alt) + e,T

1= 1

thus proving that (J cannot be an L-Nash equilibrium for arbitrarily small q,contradicting our assumption.

Indeed, W~l= wit for all t = tkm + l' k = 0, 1, 2,..., where "bad" dates do notincrease storage. Hence W~ is bounded by some ME IR~ (for example

,

M = mw). Hence

T T

~ L [uh (c~) - q W~] > ~ L U It(cD - q . M./=1' 1=1

(7.6)

Next, let t=tj andj=tkm+ 1 (k=O, 1,2,...); then (7.5) implies

8~ - C,:1

a, + ~ ill,) =I

In::. 1 (ill, + z~)-In

::.1 a,

I

m-1< (j < (j.

m

For t=tkm+I (k= 1,2,...), we have, again by (7.5) applied to all t(k-l)m+;for i = 2,..., m, k = 1, 2,...,

II

C~-(m-l

alt +~Wh )II

<:~ £, IIz~(k-l)m+i+wh-ahllm m m ;=2 '

m-l< (j < (j.

m

Therefore, by (7.4) and the monotonicity of Uh (Assumption 2),

1 T 1- L Uh(C~) > - #{t <: T

I I = II or I "bad"} Uh(O)T 1=1 T

+ ~, # {t < Tit" good"} [uh C:

1ah +

1Wh) - e]

,

.


By Lemma 3(iv), (lfT)#{t<Tlt"good"} converges to 1 as T--+oo,therefore there is To (large enough) such that

1;' ( At

(m - 1 1

)-T ~ Uh Ch) > Uh ah +-Wh - 28t=1 m m

for all T> To' Let ij be such that ij. M < e, then, by (7.6) and (7.3), wefinally have

1 ~[ At

~

t ](m - 1

.1

)( )- ~ Uh(Ch) - q . Wh > Uh ah + - Wh - 38 > Uh ah + e,T t=1 m f11.

for all T > To and all 0 4i;q < ii, completingour proof.Step 2. W(g) c [POu(q) n NEu(q)] for allq ~ O.

Let CE WG(g), and consider the stationary commodity outcome (W, z, c)with c~ = Ch, z~ = Ch- Wh' W~ = O. We can use precisely the samearguments as in the Proof of Step 2 in Theorem 2, to show that it is U-Paretooptimal and U-Nash equilibrium with respect to any q ~ O. Actually, weprove a stronger result, namely, that it cannot be dominated even in one dateT (i.e., for no T, the average payofT(s) may be higher). Also, since there is no'storage cost in (W, z, c), q does not matter.

REFE REN CE S

1. R. J. AUMANN,Acceptable points in general cooperative n-person games, Ann. Math.Studies 40 (1959), 287-324.

2. G. DEBREU AND H. SCARF, A limit theorem on the core of an economy, Internat. Eeon. .

Rev. 4 (1963), 235-246. .

3. S. HART, Lecture notes on game theory, mimeo, Institute for Mathematical Studies in theSocial Sciences, Stanford University, 1979.

4. J. R. HICKS,"Value and Capital," Oxford Univ. Press (Clarendon), London, 1939.5. L HURWICZ,Outcome functions yielding Walrasian and Lindahl al1ocations at Nash

equilibrium points, Rev. Eeon. Studies 46, No. 143 (April 1979), 217-225.6. L. HuRwlcz, On allocations attainable through Nash equilibria, J. Eeon. Theory 21,

No.1 (August 1979), 140-165.7. M. KURZ, Altruistic equilibrium, in "Economic Progress, Private Values and Public

Policy, Essays in Honor of William Fellner" (B. Balassa and R. Nelson, Eds.),Chap. 8,pp. 177-200, North-Holland, Amsterdam, 1977.

8. M. KURZ,Altruism. as an outcome of social interactions, Amer. Eeon. Rev. Papers Proe.(May 1978), 216-222.

9. M. KURZ, "A Strategic Theory of Inflation," Technical Report No. 283, Institute forMathematical Studies in the Social Sciences, Stanford University, April 1979.

10. D. LUCEANDH. RAIFFA,"Games and Decisions," Wiley, New York, 1957.

346 KURZ AND HART

1J. J. F. NASH,Two-person cooperative games, Econometrica 21 (1953), 128-140.J2. T. R. ROCKAFELLAR,"Convex Analysis," Princeton University Press, Princeton, N. J.,

1970.13. D. SCHMEIDLER,Walrasian analysis via strategic outcome functions, Econometrica 48,

No.7 (November 1980), 1585-1594.

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Pareto-Optimal Nash Equilibria Are Competitive InaRepeated … · 2004-10-12 · JOURNAL OFECONOMIC THEORY 28,320-346(1982) Pareto-Optimal Nash Equilibria Are Competitive InaRepeated